Optical metrology techniques generally referred to as scatterometry offer the potential to characterize parameters of a workpiece during a manufacturing process. In practice, light is directed onto a diffracting structure, such as a periodic grating, in a workpiece and a spectrum of reflected light is measured and analyzed to characterize unknown parameters of the diffracting structure. Characterization parameters may include critical dimensions (CD), sidewall angle (SWA), feature height (HT) and any others which vary a material's reflectivity and refractive index. Characterization of the diffracting structure may thereby characterize the workpiece as well as manufacturing process employed in the formation of the diffracting structure and the workpiece.
Analysis of a measured spectrum typically involves comparing the measurement data to theoretical spectra to deduce the parameters that best describe the measured diffracting structure. A theoretical spectrum for a set of parameter values (e.g., a line profile) can be computed using rigorous diffraction modeling algorithms, such as Rigorous Coupled Wave Analysis (RCWA). In RCWA, a model of a line profile used for predicting intensity versus wavelength is expressed as a set of stacked slices or slabs of material, as shown in FIG. 1. Each slab is defined by a width WS, a height HS, and an index of refraction nS. By means of the slicing the differential Maxwell equations can be transformed into an eigen equation. The boundary conditions of the electromagnetic field across the model profile are transformed into boundary conditions for a staircase profile. The model profile is then used as an input to a computer program that predicts the percentage of reflected energy diffracted into the zeroth order over a range of wavelengths for the model profile. The predicted intensity is compared to the measured intensity as a function of wavelength (normalized for the incident intensity) and the slab widths and heights are adjusted until agreement between the predicted and measured intensity versus wavelength curves is achieved. The final result of the analysis is a stacked set of slabs that represents the line profile of each line of the diffracting structure.
Although profiles can be approximated by model profile including a series of stacked slabs, the minimum number of slabs required for an adequate approximation with RCWA methods can be prohibitively high for certain profiles encountered in manufacturing processes. For example, structures having a relatively small SWA (i.e., shallow slope) and/or high index contrast (i.e., complex index of refraction with a large imaginary part) may suffer from poor calculation efficiency and poor convergence.